First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere

In this paper, we study the first eigenvalue of the stability operator on an integral n-varifold with constant mean curvature in the unit sphere \mathbb {S}^{n+1}. We find the optimal upper bound and prove a rigidity result characterizing the case when it is attained. This gives a new characterization for certain singular Clifford tori.